the equation of the generator to the equations of the system. Thus solving a polynomial system over a number field is reduced to solving another system over the rational numbers. For example, if a system contains , a system over the rational numbers is obtained by adding the equation and replacing by in the other equations. In the case of a finite field, the same transformation allows always supposing that the field has a prime order. Algebraic representation of the solutions Regular chains The usual way of representing the solutions is through zero-dimensional regular chains. Such a chain consists of a sequence of polynomials , , ..., such that, for every such that is a polynomial in only, which has a degree in ; the coefficient of in is a polynomial in which does not have any common zero with , ..., . To such a regular chain is associated a triangular system of equations The solutions of this system are obtained by solving the first univariate equation, substituting the solutions in the other equations, then solving the second equation which is now univariate, and so on. The definition of regular chains implies that the univariate equation obtained from has degree and thus that the system has solutions, provided that there is no multiple root in this resolution process (fundamental theorem of algebra). Every zero-dimensional system of polynomial equations is equivalent (i.e. has the same solutions) to a finite number of regular chains. Several regular chains may be needed, as it is the case for the following system which has three solutions. There are several algorithms for computing a triangular decomposition of an arbitrary polynomial system (not necessarily zero-dimensional) into regular chains (or regular semi-algebraic systems). There is also an algorithm which is specific to the zero-dimensional case and is competitive, in this case, with the direct algorithms. It consists in computing first the Gröbner basis for the graded reverse lexicographic order (grevlex), then deducing the lexicographical Gröbner basis by FGLM algorithm and finally applying the Lextriangular algorithm. This representation of the solutions are fully convenient for coefficients in a finite field. However, for rational coefficients, two aspects have to be taken care of: The output may involve huge integers which may make the computation and the use of the result problematic. To deduce the numeric values of the solutions from the output, one has to solve univariate polynomials with